Definition:Metrizable Topology

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Definition 1

$T$ is said to be metrizable if and only if there exists a metric $d$ on $S$ such that:

$\tau$ is the topology induced by $d$ on $S$.

Definition 2

$T$ is said to be metrizable if and only if there exists a metric space $M = \struct{A, d}$ such that:

$T$ is homeomorphic to the topological space $\struct{A, \tau_d}$

where $\tau_d$ is the topology induced by $d$ on $A$.

Also see

• Equivalence of Definitions of Metrizable Topology

• Indiscrete Topology is not Metrizable: thus, not all topological spaces are metrizable.

• Definition:Completely Metrizable Topology

• Results about metrizable topologies can be found here.

Linguistic Note

The British English spelling for metrizable is metrisable, but it is rarely found.

Sources

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• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces